The US state of Tennessee on Thursday used a controversial lethal injection procedure to execute a man who was convicted of raping and killing a child, after the nation's top court declined his final bid for a stay.

"I just want to say I'm really sorry. And that ... that's it," Billy Ray Irick said in his final words before prison officials in Nashville, Tennessee started the process. He was pronounced dead at 7:48 pm, officials said at a press conference.

Irick was the first inmate to be executed in Tennessee since 2009.

The US Supreme Court had earlier denied a stay of execution for the convict, rejecting concerns about the inmate potentially feeling sensations equivalent to being "burned alive."

The high court's decision was countered with a blistering dissent from Justice Sonia Sotomayor, who questioned whether Billy Ray Irick's planned execution would be too painful and whether allowing it to proceed required accepting "barbarism."

The 59-year-old was convicted in 1986 of raping and killing seven-year-old Paula Dyer. His lawyers have argued Irick has a history of severe mental illness.

In an appeal to the Supreme Court, Irick's lawyers challenged Tennessee's lethal injection protocol, which included the use of the sedative midazolam. Other states have also used the drug for executions with mixed results.

Medical experts warned the drug may not be strong enough to keep a prisoner unconscious once he starts to feel pain.

If Irick were to awaken, a paralytic used as the second drug in Tennessee's lethal injection would prevent him from alerting officials that he can feel pain, Sotomayor noted.

"Medical experts explained in painstaking detail how the three-drug cocktail Tennessee plans to inject into Irick's veins will cause him to experience sensations of drowning, suffocating, and being burned alive from the inside out," Sotomayor wrote.